Related papers: Optimal rigidity estimates for varifolds almost mi…

We consider an optical hypersurface $\Sigma$ in the cotangent bundle $\tau:T^*M\to M$ of a closed manifold $M$ endowed with a twisted symplectic structure. We show that if the characteristic foliation of $\Sigma$ is Anosov, then a smooth…

The main results of this paper provide VMO-type estimates for the quadratic tilt-excess on varifolds with critical generalized mean curvature. These estimates apply to varifolds with "almost-integral" density which are close to a…

This paper is devoted to the Moser-Trudinger-Onofri inequality on smooth compact connected Riemannian manifolds. We establish a rigidity result for the Euler-Lagrange equation and deduce an estimate of the optimal constant in the inequality…

Kuwert and Sch\"atzle showed in 2001 that the Willmore flow converges to a standard round sphere, if the initial energy is small. In this situation, we prove stability estimates for the barycenter and the quadratic moment of the surface.…

Let $(M,\theta)$ be a compact strictly pseudoconvex pseudohermitian manifold which is CR embedded into a complex space. In an earlier paper, Lin and the authors gave several sharp upper bounds for the first positive eigenvalue $\lambda_1$…

For any compact connected one-dimensional submanifold $K\subset \mathbb R^{2\times 2}$ which has no rank-one connection and is elliptic, we prove the quantitative rigidity estimate \[ \inf_{M\in K}\int_{B_{1/2}}| Du -M |^2\,dx \leq C…

For compact submanifolds in Euclidean and Spherical space forms with Ricci curvature bounded below by a function $\alpha(n,k,H,c)$ of mean curvature, we prove that the submanifold is either isometric to the Einstein Clifford torus, or a…

The study of the $k$-th elementary symmetric function of the Weyl-Schouten curvature tensor of a Riemannian metric, the so called $\sigma_k$ curvature, has produced many fruitful results in conformal geometry in recent years, especially…

We consider a $\pi_{1}$--injective immersion $f:\Sigma\to M$ from a compact surface $\Sigma$ to a hyperbolic 3--manifold $M$. Let $\Gamma$ denote the copy of $\pi_{1}\Sigma$ in $\mathrm{Isom}({\mathbb{H}}^{3})$ induced by the immersion and…

Some classification results for closed surfaces in Berger spheres are presented. On the one hand, a Willmore functional for isometrically immersed surfaces into an homogeneous space $\mathbb{E}^{3}(\kappa,\tau)$ with isometry group of…

In order to estimate roughness exponents of interface growth models, we propose the calculation of effective exponents from the roughness fluctuation (sigma) in the steady state. We compare the finite-size behavior of these exponents and…

For the Bach-flat closed manifold with positive scalar curvature, we prove a rigidity result under a given inequality involving the Weyl curvature and the traceless Ricci curvature. Moveover, under an inequality involving…

In a previous paper we developed a regularity and compactness theory in Euclidean ambient spaces for codimension 1 weakly stable CMC integral varifolds satisfying two (necessary) structural conditions. Here we generalize this theory to the…

In this article, we consider compact surfaces $\Sigma$ having constant mean curvature $H$ ($H$-surfaces) whose boundary $\Gamma=\partial\Sigma\subset \mathbb{M}_0= \mathbb{M} \times_f\{0\}$ is transversal to the slice $\mathbb{M}_0$ of the…

We prove that if $u\in C^0(B_1)$ satisfies $F(x,D^2u) \le 0$ in $B_1\subset \mathbb{R}^2$, in the viscosity sense, for some fully nonlinear $(\lambda, \Lambda)$-elliptic operator, then $u \in W^{2,\varepsilon}(B_{1/2})$, with appropriate…

In this article, we prove a rigidity theorem for isometric embeddings into the Schwarzschild manifold, by using the variational formula of quasi-local mass.

We prove that a stable $C^{1,1}$-to-edge properly embedded free boundary minimal hypersurface $\Sigma^3$ of a $4$-dimensional wedge domain $\Omega^4_{\theta}$ with angle $\theta\in (0,\pi]$ is flat.

In this paper, we show that, under arbitrary bounded Willmore energy assumption, embedded Willmore spheres (or more generally, embedded Willmore spheres under area constraint) with small diameter in a given $3$-dimensional Riemannian…

Let $(M,g)$ be an $n$-dimensional $(n\geq 3)$ compact Riemannian manifold with Ric$_{(M,g)}\geq (n-1)g$. If $(M,g)$ supports an AB-type critical Sobolev inequality with Sobolev constants close to the optimal ones corresponding to the…

For smooth mappings of the unit disc into the oriented Grassmannian manifold $\mathbb G_{n,2}$, H\'elein (2002) conjectured the global existence of Coulomb frames with bounded conformal factor provided the integral of $|\boldsymbol A|^2$,…